dx/dt=2y-x
dy/dt=e^x+y

solve y=f(x)

Question

dx/dt=2y-x
dy/dt=e^x+y

solve y=f(x)

anonymous · Answer

can we just use the fact that it is exact?

jamesj · Answer

Hm ... are you sure the second equation is e^x not e^t?

mr.math · Answer

Let me start by substituting $2y=x'+x \implies 2y'=x''+1$. We can rewrite the second DE as $x''+1=2e^x+x'+x \implies x''-x'-x=2e^x-1$.

jamesj · Answer

@imran, no you most definitely cannot do that because y is a function of x and vice versa so your integrations are incorrect.

@Mr.M: you've got it.

mr.math · Answer

But e^x is a problem.

anonymous · Answer

This is phase plane problem , btw

mr.math · Answer

I don't even know what phase plane problems are!

anonymous · Answer

b/c we can divide them
dx/dt=2y-x 
dy/dt=e^x+y 

dy/dx=  e^x+y
             -----
              2y-x

(2y-x)dy= (e^x+y) dx
 (e^x+y) dx  +(-2y+x)dy= 0
exact

e^x+yx -y^x=C